Select The Correct Answer.Factor The Polynomial Below:$\[ X^2 - 16x + 63 \\]A. \[$(x + 9)(x - 7)\$\] B. \[$(x + 9)(x + 7)\$\] C. \[$(x - 9)(x - 7)\$\] D. \[$(x - 9)(x + 7)\$\]

by ADMIN 181 views

Introduction

Factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In this article, we will focus on factoring a quadratic polynomial, which is a polynomial of degree two. We will use the given polynomial x2βˆ’16x+63x^2 - 16x + 63 as an example and guide you through the process of factoring it.

What is Factoring?

Factoring a polynomial involves expressing it as a product of two or more polynomials. This is also known as the factorization of the polynomial. Factoring is an essential tool in algebra, as it allows us to simplify complex expressions, solve equations, and identify the roots of a polynomial.

The Given Polynomial

The given polynomial is x2βˆ’16x+63x^2 - 16x + 63. This is a quadratic polynomial, which means it has a degree of two. To factor this polynomial, we need to find two binomials whose product is equal to the given polynomial.

Step 1: Identify the Factors

To factor the polynomial, we need to identify two numbers whose product is equal to the constant term (63) and whose sum is equal to the coefficient of the linear term (-16). These numbers are the roots of the polynomial.

Step 2: Use the Factoring Method

There are several factoring methods, including the difference of squares, perfect square trinomial, and factoring by grouping. In this case, we will use the factoring method that involves finding two binomials whose product is equal to the given polynomial.

Step 3: Factor the Polynomial

To factor the polynomial, we need to find two binomials whose product is equal to the given polynomial. We can start by listing the factors of the constant term (63) and identifying the pair of factors that add up to the coefficient of the linear term (-16).

The factors of 63 are: 1, 3, 7, 9, 21, and 63.

We can see that the pair of factors that add up to -16 is -9 and -7.

Therefore, we can write the polynomial as:

x2βˆ’16x+63=(xβˆ’9)(xβˆ’7)x^2 - 16x + 63 = (x - 9)(x - 7)

Conclusion

In this article, we have factored the polynomial x2βˆ’16x+63x^2 - 16x + 63 using the factoring method. We identified the factors of the constant term (63) and found the pair of factors that add up to the coefficient of the linear term (-16). We then wrote the polynomial as the product of two binomials: (xβˆ’9)(xβˆ’7)(x - 9)(x - 7).

Answer

The correct answer is:

(C) (xβˆ’9)(xβˆ’7)(x - 9)(x - 7)

Discussion

Factoring polynomials is an essential tool in algebra that involves expressing a polynomial as a product of simpler polynomials. In this article, we have used the factoring method to factor the polynomial x2βˆ’16x+63x^2 - 16x + 63. We identified the factors of the constant term (63) and found the pair of factors that add up to the coefficient of the linear term (-16). We then wrote the polynomial as the product of two binomials: (xβˆ’9)(xβˆ’7)(x - 9)(x - 7).

Additional Resources

For more information on factoring polynomials, please refer to the following resources:

  • Khan Academy: Factoring Polynomials
  • Mathway: Factoring Polynomials
  • Wolfram Alpha: Factoring Polynomials

Conclusion

Introduction

Factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In our previous article, we used the factoring method to factor the polynomial x2βˆ’16x+63x^2 - 16x + 63. In this article, we will answer some frequently asked questions about factoring polynomials.

Q: What is factoring?

A: Factoring a polynomial involves expressing it as a product of two or more polynomials. This is also known as the factorization of the polynomial.

Q: Why is factoring important?

A: Factoring is an essential tool in algebra that allows us to simplify complex expressions, solve equations, and identify the roots of a polynomial. It is also used in various fields such as physics, engineering, and economics.

Q: What are the different types of factoring?

A: There are several types of factoring, including:

  • Difference of squares: This involves factoring a polynomial of the form a2βˆ’b2a^2 - b^2.
  • Perfect square trinomial: This involves factoring a polynomial of the form a2+2ab+b2a^2 + 2ab + b^2.
  • Factoring by grouping: This involves factoring a polynomial by grouping terms together.
  • Factoring using the quadratic formula: This involves factoring a polynomial using the quadratic formula.

Q: How do I factor a polynomial?

A: To factor a polynomial, you need to identify the factors of the constant term and find the pair of factors that add up to the coefficient of the linear term. You can then write the polynomial as the product of two binomials.

Q: What are the common mistakes to avoid when factoring polynomials?

A: Some common mistakes to avoid when factoring polynomials include:

  • Not identifying the factors of the constant term: Make sure to identify the factors of the constant term before factoring the polynomial.
  • Not finding the pair of factors that add up to the coefficient of the linear term: Make sure to find the pair of factors that add up to the coefficient of the linear term before factoring the polynomial.
  • Not writing the polynomial as the product of two binomials: Make sure to write the polynomial as the product of two binomials after factoring it.

Q: How do I check if my factored polynomial is correct?

A: To check if your factored polynomial is correct, you can multiply the two binomials together and see if you get the original polynomial. If you get the original polynomial, then your factored polynomial is correct.

Q: What are some real-world applications of factoring polynomials?

A: Factoring polynomials has many real-world applications, including:

  • Physics: Factoring polynomials is used to solve problems involving motion, energy, and momentum.
  • Engineering: Factoring polynomials is used to design and optimize systems, such as bridges and buildings.
  • Economics: Factoring polynomials is used to model and analyze economic systems, such as supply and demand.

Conclusion

Factoring polynomials is an essential tool in algebra that involves expressing a polynomial as a product of simpler polynomials. In this article, we have answered some frequently asked questions about factoring polynomials. We hope that this article has been helpful in understanding the concept of factoring polynomials and its applications.

Additional Resources

For more information on factoring polynomials, please refer to the following resources:

  • Khan Academy: Factoring Polynomials
  • Mathway: Factoring Polynomials
  • Wolfram Alpha: Factoring Polynomials

Practice Problems

Try factoring the following polynomials:

  • x2+5x+6x^2 + 5x + 6
  • x2βˆ’7x+12x^2 - 7x + 12
  • x2+2xβˆ’15x^2 + 2x - 15

Answer Key

  • x2+5x+6=(x+3)(x+2)x^2 + 5x + 6 = (x + 3)(x + 2)
  • x2βˆ’7x+12=(xβˆ’3)(xβˆ’4)x^2 - 7x + 12 = (x - 3)(x - 4)
  • x2+2xβˆ’15=(x+5)(xβˆ’3)x^2 + 2x - 15 = (x + 5)(x - 3)